Self-similarity of fluvial erosion topography manifests itself through statistical renormalization properties of drainage networks and river profiles. Results based on these renormalization properties are often inconsistent with those from spectral or variogram analysis of linear transects. We show that the inconsistency is due to two factors: (1) Network and transect analyses are based on different self-similarity conditions, and (2) elevation along transects depends on the hillslopes, where self-similarity either does not apply or applies with different parameters. As a result, transect Methods produce misleading results when the topography has finite hillslopes; We also consider whether self-similarity is sufficient to describe the scaling invariance of natural river basin topography or more sophisticated multifractal conditions should be used. The conclusion of multifractality reached in recent studies was based on the use of gradient amplitudes rather than elevation increments. The gradient amplitude method is found to produce spurious multifractality also in cases when the surface is known to be self-similar. When more appropriate methods are used, topography shows little evidence of multifractality. We support our conclusions through numerical analysis of synthetic and natural river basins.